THE EXISTENCE AND UNIQUENESS OF COMPLETION OF COMPLEX VALUED METRIC SPACES

E-Jurnal Matematika Vol. 7 (2), Mei 2018, pp. 187-193

DOI: https://doi.org/10.24843/MTK.2018.v07.i02.p202

ISSN: 2303-1751

THE EXISTENCE AND UNIQUENESS OF COMPLETION OF COMPLEX VALUED METRIC SPACES

Dahliatul Hasanah^1§, Imam Supeno²

¹Jurusan Matematika, Fakultas MIPA – Universitas Negeri Malang [Email: dahliatul.hasanah.fmipa@um.ac.id] ²Jurusan Matematika, Fakultas MIPA – Universitas Negeri Malang [Email: imam.supeno.fmipa@um.ac.id] ^§Corresponding Author

ABSTRACT

Completion of a metric space has been investigated and proved its existence and uniqueness. While a complex valued metric space is considered as a generalization of a metric space, we study the completion properties of complex valued metric spaces. Using redefined definitions of isometry and denseness in complex valued metric spaces which correspond to the definitions in metric spaces, we prove the existence and uniqueness of completion of a complex valued metric space.

Keywords: complex valued metric spaces, completion, isometry, denseness

• 1.    INRODUCTION

Complex valued metric spaces are considered as a generalization of the classical metric spaces by redefining the metric. It was introduced by Azzam et.al (2011) to investigate the existence of common fixed points of a pair of mappings satisfying contractive type condition in the new spaces. Following the definition of complex valued metric spaces, many researchers started to investigate other fixed point theorems in the spaces such as in Sintunavarat & Kumam (2012), Sitthikul, et. Al. (2012), and Nashine, et. al. (2014).

A well-known property of metric spaces is the existence and uniqueness of completion of a metric space. In other words, every metric space has a completion. Completion of other type of metric spaces is also studied in recent years. For example, Ge & Lin (2015) investigate the existence of completion of partial metric spaces and Dung (2017) provides an example of completion of partial metric spaces as an answer to Ge & Lin’s question about denseness property. On the other hand, Cevik & Okezen (2016) study the completion of multiplicative metric spaces. Completion of quasipseudometric spaces is investigated by

Andrikopoulos (2013). It is still an open problem whether a complex valued metric space has a completion. To investigate the existence and uniqueness of completion, we introduce some notions in complex valued metric spaces such as isometry and denseness. The definitions of new notions correspond to the same notions in metric spaces.

• 2.    PRELIMINARIES

In this section we give some preliminaries on complex valued metric spaces and some basic definitions and properties in the spaces.

Definition 2.1. (Azzam, et.al., 2011) A partial order ^ on C is defined as follows.

^zι ~ z₂ if and only if Rzz^zι ) ≤ R z(^z2 ) and Im(z₁) ≤ Imfz₂ ).

From the definition, it follows that z₁ ^ z₂ if one of the following conditions is satisfied:

• i.    Re(z₁ ) = Rz(z₂), Im(z₁) < Im(z₂) ii. Re(z₁) < Re(z₂I, Zm(z= = Im(Zf) iii. Rz(zf) < Re(z₂I, Imfz-f) < ImZzf) iv. Rz(zf) = Re(z₂), Zm(z= = Imfz₂)

The notation z₁ ⅛ z₂ is used only if the condition (i), (ii), or (iii) is satisfied. If the condition (iii) is satisfied, then it is written as ^zι < ^Z2-

The properties of the order ^ are given by

• a)    0 < Z^ ^ Z2 implies ∖z₁∖ < ∖z₂∖

• b)    z₁ ^ z₂ and z₂ < z₃ implies z₁ < z 3^.

Definition 2.2. (Azzam, et.al., 2011) Let X be a nonempty set. Suppose that for all x, y, z ∈ X, the mapping d∙. X × X → C satisfies

(C1) 0 ^ d(x,y) and d(x,y) = 0 if and only if x — y;

(C2) d(x,y) — d(y,x);

(C3) d(x,y) ^ d(x, z) + d(z,y).

Then d is called a complex valued metric on X, and (X, d) is called a complex valued metric space.

The followings are given definition of interior and limit point in complex valued metric spaces and definition of open and closed sets in the spaces.

A point x ∈ X is called interior point of a set AXX whenever there exists 0 < r ∈ C such that

B(x, r) — {y ∈ X: d(x, y) <r} X A.

A point x ∈ X is called a limit point of A whenever for every 0 < r ∈ C,

B(x,r)∩(A∖{x})≠0.

A subset AXX is called open whenever each element of A is an interior point of A. A subset XXX is called closed whenever each limit point of B belongs to B.

Convergence of a sequence and a Cauchy sequence in complex valued metric spaces are defined as follows.

Let (x_n) be a sequence in X and x ∈ X. If for every c ∈ C, with 0 < c, there is n₀ ∈ N such that for all n > n₀, d(x, x_n) < c, then (x_n) is said to be convergent. Sequence (x_n) converges to x and x is said to be the limit point of (%_n). This convergence is denoted by lim_n→∞ x_n — x or x_n → x as n → ∞. If for every c ∈ C, with 0 < c, there is n₀ ∈ N such that for all n > n₀, d(x_n,x_n+m) < c, then (x_n) is called a Cauchy sequence in (X, d). If every Cauchy sequence is convergent in (X, d) then (X, d) is called a complete complex valued metric space.

The following are lemmas related to properties of sequences in complex valued metric spaces.

Lemma 2.3. (Azzam, et.al., 2011) Let (X, d) be a complex valued metric space and let (x_n) be a sequence in X. Then (x_n) converges to x if and only if ∖d(x_n,x)∖ → 0 as n → ∞.

Lemma 2.4. (Azzam, et.al., 2011) Let (X, d) be a complex valued metric space and let (x_n) be a sequence in X. Then (x_n) is a Cauchy sequence if and only if ∖d(x_n,x_n+m)∖ → 0 as n → ∞.

Lemma 2.5. Let (X, d) be a complex valued metric space. Let (x_n) and (y_n) be sequences in X such that (x_n) → x and (y_n) → y. Then d(xn,yn) → d(x,y).

PROOF. By the property of complex valued metric, we have

d(x_n,y_n) ^ d(x_n,x) + d(x,y) + d(y,y_n). Taking the absolute value and limit of both sides, we get

lim∖d(x_n,y_n)∖ ≤ lim∖d(x_n,x)∖ +lim∖d(x,y)∖ + lim∖d(y,yn)∖ — lim∖d(x,y)∖ — ∖d(x,y)∖.

On the other hand,

d⁽x,y) < d⁽xn,x) + d(xn,yn) + d⁽y,yn). Taking the absolute values and limit of both sides, we have

∖d(x,y)∖ ≤ lim∖d(xn,yn)∖.

Hence we have lim∖d(x_n,y_n)∖ — ∖d(x,y)∖ or we can say that d(x_n,y_n) → d(x,y).  □

The followings are the definitions of isometry and denseness in a complex valued metric space by adopting the description of isometry and denseness for partial metric case in Ge & Lin (2014).

Definition 2.6. Let (X, d') and (Y, p) be complex valued metric spaces. A mapping f:X → Y is called an isometry if pff(x),f(y)) — d(x,y) for all x,y ∈ X.

The following lemma states the relation between isometry and convergence of a sequence.

Lemma 2.7. Let f:X → Y be an isometry, where (X,d) and (Y, p) are complex valued metric spaces. if (x_n) is a sequence in X which

converges to , then ( (  )) is a sequence in

converging to ( ).

PROOF. Suppose that (  ) is in and →

. We will show that ( (  )) converges to

().

For any 0≺  ∈ℂ, there exists ∈ℕ

such that for all >   ,  (  , )≺ . On the

other hand,  ( (  ),  ( ))=  (  , ). Hence

( (  ),  ( ))≺  . This means that ( ())

converges to f(x).□

Definition 2.8. A subset of a complex valued _ metric space is said to be dense in if =

Form the definition above, taking an element of would yield an element of itself or a limit point of . Thus this means that a subset of a complex valued metric space is called dense in if for each element of , there exist an element of in every neighborhood of the element of .

Other that denseness in Definition 2.8, we introduce the notion of sequentially dense in complex valued metric spaces as the same notion that was introduced by Ge and Lin (2015) in partial metric spaces.

Definition 2.9. Let ( , ) be a complex valued

metric space and be a subspace of . is called sequentially dense in if for any ∈ there is a sequence in converging to .

There is a equivalent relation between the notion of dense set and sequentially dense set in complex valued metric spaces as stated in the Lemma as follows.

Lemma 2.10. Let be a complex valued metric space. Let be a subset of . The set is dense in if and only if it is sequentially dense in .

PROOF. Let be a dense subset of and

∈  . Then is in or is a limit point of

. Suppose that is in . Then the sequence ( , , ,…) in converges to . If is not in

then is a limit point of .

Taking  =1+ ,0≺  , there exists ∈

such that ∈  ( ,  )∩(  ∖{ }).

11

=  +   ,0≺  , there exists

∈ such that    ∈  ( ,  )∩(

and ≠.

Using similar fashion, for every =+

,∀ ∈ℕ, there exists ∈ such that

∈  ( ,  )∩(  ∖{ }).

We have a sequence (  ,  ,  ,…) in such

that

(  , )≺  =  +  ,∀

Tl Tl

We will show that (  )→.

For any =   +   ,  ,  ∈ℝ,0≺ , using

the Archimedean property, there exist

11

, ∈ℕ such that <    and <.

11

Hence +   ≺  +.

n₁n

Taking     = max{ ,  } so that for all

>,

(  , )≺  =  +   ≾+

Tl Tl Tl₀ Tl₀

≾   +    ≺  +   =.

n₁ τι₂

So (  ) converges to . Thus for every in

there exists a sequence in     which is

converging to . is sequentially dense in . On the other hand, suppose thatis

sequentially dense in . We will show that every in is in ^̅.

If ∈ then ∈ ^̅. Suppose that ∉,

there is (  ) in such that (  )→  . We

will show that is a limit point of . For any 0≺ ∈ℂ, there is ∈ℕ such that for all >   ,   (  , )≺ . Take     ∈ where

>    then     ≠ and     ∈  ( , )∩

(  ∖{ }). So is a limit point of . Hence

sequentially denseness implies denseness.

The following definition gives the basic concept of completion in the sense of complex valued metric spaces.

Definition 2.11. Let ( , ) be a complex valued

metric space. A complete complex valued metric space ( ^∗,  ^∗) is called a completion of ( ,  ) if

there exists an isometry :   →  ^∗ such that

( ) is dense in ^∗.

• 3.    RESULTS AND DISCUSSION

In this section we give the theorems of existence and uniqueness of completion for complex valued metric spaces. Theorem 3.1 states the existence of the completion, while the uniqueness of the completion is mentioned in Theorem 3.2. To prove the existence of completion, some steps must be taken. The first step is to construct a complex valued metric spaces. The next step is to construct an isometry based on Definition 2.6 from X to some subset of the costructed complex valued metric space. The next step that follows is to prove that the subset is dense based on the Definition 2.9. The final step is to prove that the complex valued metric space is complete.

Theorem 3.1. Every complex valued metric space has a completion.

Proof. Let (X, d') be a complex valued metric space. Define a set C as

C = {(^χ_n): (^χn) is Cauchy sequence in X} We define a relation ~ on C as follows: For (Xn) and (y∏^) in C, (^χn) ~ (Vn) ^

lim∣d(xn,yn)∣ = 0.

We will show that the relation ~ is an

equivalence relation.

• (i)    Reflexive

Note that for (x_n) ∈ C, based on the property complex valued metric, d(x_n,x_n) = 0,Vn∈ N.            Hence

lim∣d(x_n,y_n)∣ = 0. So (x_n) ~ (x_n).

• (ii)    Symmetry

Suppose that (x_n), (y_n) ∈ C satisfy (x_n) ~ (y_n). Based on the property of complex         valued         metric,

O = lim∣d(xn,yJ^l=lim^l⅜n,xn)^l∙ Thus we have (y_n) ~ (x_n).

• (iii)    Transitive

Suppose that (  ), (  ), (  ) ∈ satisfy

(^χn) ~ (Xn) and ⁽yn) ~ ⅛). We have, d(x_n,y_n) ~ d(x_n, z_n) + d(z_n, y_n).

By the property of partial order ^, ∣d(xn,Zn)∣ ≤ ∣d(xn,yn) + d(y_n,Z_n)∣

≤ ∣d(xn,yn)∣ + ∣d(yn,Zn)∣.

Taking limit both sides yields

lim∣ (   ,   )∣

≤ lim∣ (  ,   )∣

+ lim∣d(y_n,z_n)∣ = 0 + 0

= 0.

Thus lim∣ (  ,  )∣ = 0 and hence

(Xn) ~ ⁽Zn).

So, the relation ~ is an equivalence relation.

Let X* be the set of all equivalence classes in C for the equivalence relation ~, that is X* = {[(^χ_n)]: (^χ_n) ∈ C}. We write (x'_n) ∈ [(^χ_n)] to mean that (x'_n) ~ (x_n) and (x'_n) is a representative of the equivalence class [(  )].

We now define d*∙.X*×X*→<C as follows: for [(Xn)],[(Xn)]∈X*,

d *([(^χn)],[(Xn)]) = limd(x_n,y_n).

We will show that d* is well-defined by showing that limd(x_n,y_n) exists for arbitrary Cauchy sequences (x_n) and (y_n) in X and the limit is independent of the particular choice of the representatives of the equivalence classes.

• (i)    For Cauchy sequences (x_n) and (y_n) in X, d(xn,yn) ⅛ d(x_n, X_n+m) + d Cx_n+m, y n+m)

+ dCy_n+m> Vn)

d(xn,yn) - dCx_n+m, y_n+m)

≤ dCx_n, x_n+m)

+ d(yn₊m> y_n)

^{∣ ( , ) - ( ,)}

< ∣d(xn,xn+m)∣

^{+ ∣ ( , )∣}

By taking limit on both sides we have

lim∣ (   ,   ) -  (      ,)

= lim∣ (  ,)

+ lim∣d(yn_+m>yn)∖-

From Lemma 2, we have

lim∣ (      ,   )∣ = lim∣ (      ,   )∣ = 0.

Hence, lim∣ (  ,   ) -  (     ,      )∣ = 0.

This means that (  ,  ) is Cauchy

sequence in C. Since C is a complete then d (x_n,y_n)    converges in   C,   thus

lim (  ,   ) does exist.

• (ii)    Let (  ), (  ), (  ), (  ) be in   such that

(xn) ~ ⅛)  and (yn) ~ ⁽y_n).  We will

show that the limit is independent of the particular choice of the representatives of the equivalence classes by showing that lim (   ,   ) = lim (   ,   ).

By the property of complex valued metric and the equivalence relation,

d(xn,yn) ⅛ dCx_n, x)) + dCx_n, y))

+ dCyn, y_n)

d(χ_n,y_n) - d(xn, y)ι)

• < dXx_n, x'_n} + dyyn, y_n) lim∣d(x_n,y_n) - dx×h,y))∖

≤ lim∖d(x_n,xh)∖

+ lim∖d(yn,yn)∖ = 0.

This would imply   lim[d(x_n,y_n)—

d (xⁿ,yn)] = 0.

Since dXx_n, y_n) is convergent sequence in C, then d(Xn, y)) is also convergent. Thus lim d(x'_n, y)} does exist and

limd(x_n,y_n) = limd(xl_l,y)).

Consequently,  d*∙. X*× X*→C is well-

defined.

We will show that d* is complex valued metric on ^*.

• (i)    Let   ^[(xn^)],^[(yn^)] ∈ X*.   We have

d*([(Xn)], [(y∏)]) = lim d(x_n, y_n).    Since

0 < d(x_n,y_n),   then 0 < lim d(x_n, y_n).

Hence d* satisfies the first axiom of complex valued metric.

• (ii)    Let [(x_n)], [(y_n)] ∈ X*. Note that using the property of complex valued metric we                            have

d*([(x_n)], [(.X_n)]) = lim d(x_n, x_n) = 0. On the other hand, if we have 0 = d*^([(Xn^)], ^[(yn^)]) = limd{x_n, y_n), then this implies lim∖d(x_n,y_n)∖ = 0. This would mean that (x_n)~(y_n). Hence [(x_n)] = [(yn)l

• (iii)    For [(Xn)], [(y_n)] ∈ X*, d*^([(Xn^)], [^n)]) = limd(xn,yj = limd(y_n, x_n) = d*X[Xy_n)], [(*n)]).

• (iv)    d*([(%n)], [(z∏)]) = lim d(x_n, z_n) < limd(x_n,y_n) + lim d(y_n, z_n) = d*([(xn)l[(yn)]) + d*([(yn)],[(Zn)]).

The property of triangle inequality completes the proof that d* is complex valued metric on X*.

The next step is to construct an isometry from X to X* based on Definition 2.6. For each x ∈ X, let x* be the equivalence class of the constant sequence (x,x,x,^), that is x* = [(x,x,x,...)] ∈ X*. We define f-X →X* by f(x) = ^χ*- We will show that f is an isometry.

For x,y ∈ X,

d*ff XxXfXy)") = d*xx*,y*) = limd(x,y) = d(x,y).

Based on Definition 2.6, f-X→X* is an isometry.

We will show that f(X) is dense in X*. We consider any x* ∈ X*. Let (x_n) ∈ x*. Thus (x_n) is Cauchy sequence in X, that is ∀0 < c ∈ C,∃n₀ ∈ N such that dxx_n,x_n+m} <

c for all n>n₀. Let (x_+o,x_+o,x_+o, ...) ∈ [(⅞_l)∖. Then we have [(^xn₀)l ∈f^). From the definition of d* we have

d*([(xno)∖,x*) = d*([(xno)∖,[X%n)]) = limd(x_no,x_n).

Since d(x_no,x_n) < c,∀n> n₀, then we have limdxx_no, x_n) < c.

So we get ^d*([(xn₀)∖,x*) = limd(xn₀,xn) < c ,∀0 < c ∈ C. This means that every neighborhood of arbitrary x* ∈ X* contains an element of f(X). Thus by Definition 2.8, f (X) is dense in X*.

The last step is to show that X* is a complete complex valued metric space, that is every Cauchy sequence in X* is convergent in X*. Let (x*) be any Cauchy sequence in X*. Since f(X) is dense in X*, then for every x*, there is y* ∈ f(X) such that

d*(x*,y*) <- + i-. n n

This implies lim∖ d*(x*,y*)∖ = 0.

We will show that X ^*) is also a Cauchy sequence in ^*. By the triangle inequality of complex valued metric, we get

d*(y*,y*_+m) 3 d* Xy*, x*_n) + d* xx*_n, x*+_+m) + d*(x*_+m,y^_+m)-

Taking modulus of each side,

∖d*(yZy*₊m)∖ ≤ ∖d*Xy*,x*)∖

+ ∖  ^*X ^*,  ^*    )∖

+ ∖d*Xx*_+m,y*_n+m)∖.

Then, lim∣d*Xy*,y*₊m)l = 0.

Hence X ^*) is Cauchy sequence in X ). Suppose that y_n ∈ X such that fXy_n) = y*,∀n. We will prove that X ) is a Cauchy sequence in . Note that

d *Xy*,y*+m) = d*(jXy_n), fXy+_+m)) = d^yy_n, n++m).

Taking modulus and limit on both sides, we have

lim∖dXy_n, y _n+m)| = lim∖d*Xy*,y*+m)| = 0.

Hence, X ) is Cauchy sequence in .

Suppose that ^* ∈ ^* is the equivalence class in ^* in which X ) belongs. We will show that X ^* ) is converging to ^*. By the triangle inequality,

d*Xx*,x*) < dX x×Zy)) + d* Xy*, x*).

For fixed , the constant sequence X ,  ,  ,.) belongs to ^* and the sequence

Xy_m) is in x*. So d*Xy*,x*) = limXyn,ym). Thus we have

d*Xx*_n,x*) < d*Xx*,y₇*) + limXyn,ym).

Since we have lim| ^∗( ^∗, ^∗)|=0, taking

modulus and limit on both sides yields

lim ^∗( ^∗, ^∗)=0.

This implies ( ^∗) converges to ^∗. Hence ( ^∗, ^∗) is complete complex valued metric

space.

This completes the proof that every complex valued metric space has a completion. □

Theorem 3.2. The completion of a complex valued metric space ( , ) is unique with respect

to isometry under denseness. More precisely, if ( ^∗, ^∗) and ( ^̃, ^̃) are two completions of ( ,  ), then there is a unique isometry from ^∗

to ^̃.

PROOF. Suppose that ( ^∗, ^∗) and ( ̃, ̃) are two completions of ( , ). Let :  → ^∗ and

:  → ^̃ be the isometries by which ^∗ and ^̃

are completions of . Note that the inverse of , that is , is also an isometry. Consider the mapping ∘    : ( )→  ( ) where

( )⊆  ^∗ and ( )⊆ ^̃. Now we will define

a mapping ℎ: ^∗→ ^̃ such that the mapping is an isometry. Define the mapping ℎ as follows.

• •  If  ^∗∈  ^∗ and  ^∗ ∈  ( ), then ℎ( ^∗)=

(  ∘     )( ^∗)

• •   If   ^∗ ∈  ^∗ and  ^∗ ∉  ( ), we have ( )

is dense in ^∗. Based on Lemma 2.10, ( ) is sequentially dense in ^∗. Thus there exists a sequence ( ^∗)⊆  ( ) such

that ( ^∗)→  ^∗. Since ( ^∗)⊆  ( ), then

there is ∈ such that (  )=  ^∗ for

all  ∈ℕ. Since ( ^∗) is Cauchy sequence

in ^∗ and is an isometry then (  ) is

also a Cauchy sequence in X. By the isometry     :  → ^̃, the sequence

( (  )) =(̃) is also a Cauchy

sequence in ^̃. By the completeness of ^̃, ( ̃) is converging to some ̃ in ^̃. Put ℎ( ^∗)=̃.

We will show that the mapping ℎ: ^∗→ ^̃ is well-defined.

For ^∗, ^∗∈  ( )⊆  ^∗ and ^∗=  ^∗, ℎ( ^∗)=

( ∘    )( ^∗)=( ∘    )( ^∗)=ℎ( ^∗) since

and are isometries. Now we consider ^∗ ∈  ^∗ and   ^∗ ∉ ( ).  The set   ( )is

sequentially dense in ^∗, hence we suppose that ( ^∗) and ( ^∗) are sequences in   ()

which are converging to ^∗. For all , we have  ^∗ and  ^∗ are in  ( ) so ℎ( ^∗)and

ℎ( ^∗) are in ^̃. By the isometries of  and,

(ℎ( ^∗)) and (ℎ( ^∗)) are Cauchy sequences and thus are convergent in ^̃. Note that

̃(ℎ( ^∗),ℎ( ^∗))

= ̃((  ∘    )( ^∗),(  ∘    )( ^∗))

= ̃( (    ( ^∗)),  (    ( ^∗)))

=  (    ( ^∗),(

=  ^∗( (    ( ^∗)),  (    ( ^∗)))

=   ^∗( ^∗,

Since ( ^∗) and ( ^∗) converge to  ^∗, so we

have

| ̃(ℎ( ^∗),ℎ( ^∗))|=| ^∗( ^∗,

≤|  ^∗( ^∗,  ^∗)|+|  ^∗( ^∗,

→0.

Suppose that (ℎ( ^∗))→ and (ℎ( ^∗))→,

then we get

|  ^∗( ,  )|≤| ^∗( ,ℎ( ^∗))|

+| ^∗(ℎ( ^∗),ℎ( ^∗))|

+| ^∗(ℎ( ^∗),)|=0.

By the properties of complex valued metric, =  . So (ℎ( ^∗)) and (ℎ( ^∗)) are

converging to the same limit. Thus the mapping ℎ: ^∗→ ^̃ is well-defined.

The next step is to show that the mapping is an isometry. Let ^∗, ^∗ ∈ ^∗. Then there are sequences ( ^∗) and ( ^∗) in ( ) which converge to ^∗ and ^∗ respectively. By Lemma 2.7, (ℎ( ^∗))→ℎ( ^∗) and (ℎ( ^∗))→ ℎ( ^∗). This means

^̃(ℎ( ^∗), ℎ( ^∗))

=lim ^̃(ℎ( ^∗),ℎ( ^∗))

=lim ^̃((  ∘    )( ^∗),(  ∘    )( ^∗))

=lim ^∗( ^∗,  ^∗)

=   ^∗( ^∗,  ^∗).

So the mapping ℎ: ^∗ → ^̃ is an isometry.

To prove the uniqueness of the isometry, we will show that if there exists other isometry from ^∗ to ^̃, then they are the same. Let

:  ^∗→ ^̃ be an isometry. We will show that

ℎ= .

Let ^∗ ∈  ^∗, then there is a sequence ( ^∗)⊆

( ) which is converging to ^∗. The mapping is an isometry, then

| ̃( ( ^∗), ( ^∗))|=| ^∗( ^∗,  ^∗)|→0.

So ( ( ^∗)) converges to ( ^∗). On the other hand, ( ( ^∗)) is also converging to ℎ( ^∗). By the uniqueness of limit of sequence, we have ( ^∗)=ℎ( ^∗).

We complete the proof that the completion of a complex-valued metric space is unique. □

4. CONCLUSION AND SUGGESTION

In this paper, we propose the definitions of isometry and denseness in complex valued metric spaces based on those in metric spaces. The definitions are used to prove the existence and uniqueness of complex valued metric spaces. Hence complex valued metric spaces follow metric spaces in term of the property of having completion for each metric space.

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